We consider universal computability of the LCTRS with only rule scheme Calc:

  Signature: eval_0 :: Int -> Int -> Int -> o
             eval_1 :: Int -> Int -> Int -> o

  Rules: eval_0(x, y, z) -> eval_1(x, y, z) | y > 0 /\ x = x /\ z = z
         eval_1(x, y, z) -> eval_1(x + y, y, z) | y > x /\ z > y /\ y > 0
         eval_1(x, y, z) -> eval_1(x, y, x - y) | y > x /\ z > y /\ y > 0

The system is accessible function passing by a sort ordering that equates all sorts.

We start by computing the initial DP problem D1 = (P1, R UNION R_?, i, c), where:

  P1. (1) eval_0#(x, y, z) => eval_1#(x, y, z) | y > 0 /\ x = x /\ z = z
      (2) eval_1#(x, y, z) => eval_1#(x + y, y, z) | y > x /\ z > y /\ y > 0
      (3) eval_1#(x, y, z) => eval_1#(x, y, x - y) | y > x /\ z > y /\ y > 0

***** We apply the Graph Processor on D1 = (P1, R UNION R_?, i, c).

We compute a graph approximation with the following edges:

  1: 2 3 
  2: 2 3 
  3:

There is only one SCC, so all DPs not inside the SCC can be removed.

Processor output: { D2 = (P2, R UNION R_?, i, c) }, where:

  P2. (1) eval_1#(x, y, z) => eval_1#(x + y, y, z) | y > x /\ z > y /\ y > 0

***** We apply the Integer Function Processor on D2 = (P2, R UNION R_?, i, c).

We use the following integer mapping:

  J(eval_1#) = arg_2 - arg_1 - 1

We thus have:

  (1) y > x /\ z > y /\ y > 0 |= y - x - 1 > y - (x + y) - 1 (and y - x - 1 >= 0)

All DPs are strictly oriented, and may be removed.  Hence, this DP problem is finite.

Processor output: { }.